Lehel's approximate conjecture for uniform tight cycles

For $k\geq3$, let a $k$-uniform tight cycle be a cyclic sequence of vertices whose edges are the sets of $k$ consecutive vertices. Consider a red-blue edge-colouring of the complete $k$-graph on $n$ vertices.

Approximate Lehel conjecture. There exist constants $c_k\geq1$ and $n_0(k)\in\mathbb N$ such that, for every $n\geq n_0(k)$, the coloured complete $k$-graph contains vertex-disjoint red and blue tight cycles whose union contains at least $n-c_k$ vertices.

The paper proves the weaker bound $n-o(n)$ and notes that the error cannot be zero. A constant error is known for $k=3$, while the conjecture remains open for every $k\geq4$.